﻿390 
  Mr. 
  R. 
  H. 
  M. 
  Bosanquet 
  on 
  the 
  

  

  Plateau 
  (J.), 
  For. 
  Mem. 
  E.S. 
  Sur 
  les 
  Couleurs 
  accidentelles 
  ou 
  subjec- 
  

   tives. 
  8vo. 
  Bruxellesl875. 
  The 
  Author. 
  

  

  Smee 
  (Alfred), 
  F.K.S. 
  The 
  Mind 
  of 
  Man 
  : 
  being 
  a 
  Natural 
  System 
  of 
  

   Mental 
  Philosophy. 
  8vo. 
  London 
  1875. 
  The 
  Author. 
  

  

  The 
  Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave, 
  and 
  the 
  Practical 
  

   Treatment 
  of 
  the 
  Musical 
  Systems 
  thus 
  obtained. 
  Revised 
  

   Version 
  of 
  a 
  Paper 
  entitled 
  ( 
  On 
  Just 
  Intonation 
  in 
  Music 
  ; 
  

   with 
  a 
  description 
  of 
  a 
  new 
  Instrument 
  for 
  the 
  easy 
  control 
  of 
  

   Systems 
  of 
  Tuning 
  other 
  than 
  the 
  equal 
  Temperament 
  of 
  12 
  

   Divisions 
  in 
  the 
  Octave. 
  By 
  R. 
  H. 
  M. 
  Bosanquet, 
  Fellow 
  of 
  

   St. 
  John's 
  College, 
  Oxford. 
  Communicated 
  by 
  H. 
  J. 
  Stephen 
  

   Smith, 
  F.R.S., 
  Savilian 
  Professor 
  of 
  Geometry 
  in 
  the 
  University 
  

   of 
  Oxford. 
  Received 
  Dec. 
  4, 
  1872. 
  Read 
  Jan. 
  30, 
  1873*/ 
  33 
  

   Received 
  November 
  24, 
  1874. 
  

  

  Contents. 
  

  

  Page 
  

  

  Mode 
  of 
  expressing 
  intervals 
  in 
  E. 
  T. 
  

  

  semitones 
  390 
  

  

  Definitions 
  391 
  

  

  Intervals 
  formed 
  by 
  fifths 
  392 
  

  

  Regular 
  systems. 
  Theorems 
  a, 
  /3 
  .. 
  . 
  392 
  

   Regular 
  cyclical 
  systems. 
  Th. 
  i. 
  ii., 
  iii. 
  393 
  

  

  Th. 
  iv 
  394 
  

  

  Multiple 
  systems. 
  Th. 
  v 
  395 
  

  

  Formation 
  of 
  major 
  thirds 
  in 
  positive 
  

  

  and 
  negative 
  systems 
  396 
  

  

  Notation 
  for 
  positive 
  regular 
  systems 
  396 
  

   Notation 
  applicable 
  to 
  all 
  regular 
  

  

  systems, 
  negative 
  as 
  well 
  as 
  positive 
  397 
  

   Formation 
  of 
  harmonic 
  sevenths 
  in 
  

  

  positive 
  and 
  negative 
  systems 
  397 
  

  

  Concords 
  of 
  regular 
  and 
  regular 
  cy- 
  

   clical 
  systems 
  398 
  

  

  In 
  regular 
  cyclical 
  systems, 
  to 
  find 
  

   the 
  number 
  of 
  units 
  in 
  any 
  interval 
  

  

  in 
  the 
  scale 
  399 
  

  

  Employment 
  of 
  positive 
  systems 
  in 
  

  

  music. 
  Rule 
  for 
  thirds 
  400 
  

  

  Use 
  of 
  the 
  notation 
  with 
  musical 
  

  

  symbols 
  400 
  

  

  Principle 
  of 
  symmetrical 
  arrange- 
  

  

  Page 
  

  

  ment 
  in 
  regular 
  systems. 
  Positive 
  

   and 
  negative 
  systems. 
  401 
  

  

  Application 
  of 
  principle 
  of 
  symme- 
  

   trical 
  arrangement 
  to 
  a 
  " 
  gene- 
  

   ralized 
  key-board 
  " 
  for 
  regular 
  

   systems 
  403 
  

  

  Application 
  of 
  the 
  positive 
  system 
  of 
  

   , 
  perfect 
  thirds 
  to 
  the 
  "generalized 
  

   key 
  -board" 
  (Helmholtz's 
  system, 
  

   just 
  intonation) 
  403 
  

  

  Application 
  of 
  the 
  notation 
  of 
  posi- 
  

   tive 
  systems 
  to 
  the 
  system 
  of 
  53... 
  404 
  

  

  Application 
  of 
  the 
  system 
  of 
  53 
  to 
  

  

  - 
  the 
  "generalized 
  key-board" 
  405 
  

  

  Appli 
  cation 
  of 
  the 
  system 
  of 
  118 
  to 
  

   the 
  " 
  generalized 
  key-board 
  " 
  405 
  

  

  Application 
  of 
  the 
  negative 
  system 
  of 
  

   perfect 
  thirds 
  (mean-tone 
  system) 
  

   to 
  the 
  "generalized 
  key-board"... 
  406 
  

  

  Application 
  of 
  the 
  negative 
  system 
  of 
  

   31 
  to 
  the 
  "generalized 
  key-board" 
  406 
  

  

  The 
  investigation 
  of 
  cycles 
  of 
  the 
  

   higher 
  orders 
  — 
  the 
  new 
  cycle 
  of 
  

   643 
  and 
  others 
  406 
  

  

  The 
  mode 
  of 
  expressing 
  Intervals. 
  

  

  In 
  the 
  original 
  paper 
  presented 
  by 
  the 
  writer 
  to 
  the 
  Royal 
  Society, 
  

   logarithms 
  were 
  employed 
  as 
  the 
  measure 
  of 
  intervals, 
  as 
  they 
  have 
  been 
  

  

  * 
  See 
  1 
  Proceedings, 
  ' 
  vol, 
  xxi, 
  p. 
  131, 
  

  

  